Families of exotic A^1-contractible affine threefolds

Algebro-Geometric and Homotopical Methods

14 February 15:00 - 16:00

Adrien Dubouloz - Université de Bourgogne

After the work of Asok and Doran who constructed many examples of A^1-contractible quasi-affine varieties in every dimension higher than or equal to 4, the question of existence and abundance of "exotic" A^1-contractible affine varieties, i.e. A^1-contractible affine varieties that are not isomorphic to an affine space, remained open. An important step in this direction was made by Hoyois, Krishna and Østvær who proved that the so-called Koras-Russell threefolds, a family of topologically contractible rational smooth complex affine threefolds non-isomorphic to the affine 3-space become A^1-contractible after some finite suspension with the pointed projective line. In this tak, I will explain how deduce the A^1-contractibilty of a large class of affine threefolds non isomorphic to the affine 3-space, including Koras-Russell threefolds of the first kind, from additional geometric input related to additive group actions on such varieties. I will also review some perspective on the construction of A^1-contractible affine varieties of higher dimension. (Joint work with Jean Fasel, Université Grenoble-Alpes).